But my knowledge of these topic remains too abstract to digest it well. So i am collecting enlightening toy examples. For example, I've worked with the Legendre family of elliptic curves

{$y^2=x(x-1)(x-\lambda)$} $ \to $ {$\mathbb C -(0,1)$}

and interpreted everything into a concrete term.(and it was fantastic)

But i still wants more. Because in my examples, no mixed Hodge structure, no Hodge structure of weight $\ge$ 2. If you have any other good examples, please tell me. Good reference will be extremly helpful. I also appreciate any suggestion.

2 Answers
2

I guess the implied question is: what are good references containing explicit calculations of variations of Hodge structure etc.? I might suggest taking a look at Griffiths' early pioneering papers "On periods of certain rational integrals I, II" Annals 1969, and
"Periods of integrals on algebraic manifolds III" IHES 1970. These papers contain a large number of explicit calculations on VHS and intermediate Jacobians for things like hypersurfaces in projective space. Regarding
(variations of) mixed Hodge structures, take a look at the books by Carlson-Müller Stach-Peters, Peters-Steenbrink, and Voisin.

Carlson-Muller Stach-Peters is simply great. Full of motivation, inspiring pictures and detailed explaination. They treated my toy example in more depth! It would be a great companion of Voisin's book! Thank you so much.
–
ChoaApr 27 '12 at 14:28

Griffiths' language is slightly old, but they are great too.
–
ChoaApr 27 '12 at 14:31

In the same direction, some nice examples come from Teichmuller curves. You can check out e.g. the two preprints by Alex Wright:
http://arxiv.org/abs/1203.2683
("Schwarz triangle mappings and Teichmüller curves I: abelian square-tiled surfaces")
and
http://arxiv.org/abs/1203.2685
("Schwarz triangle mappings and Teichmüller curves II: the Veech-Ward-Bouw-Möller curves").

Also, some shameless self-advertising: there is http://arxiv.org/abs/1112.5872
by M. Kontsevich, A. Zorich and me (about square-tiled surfaces) which is mostly expository.